Smithfield market is a really interesting place, and if you can believe it smack dab in the middle of the market is a fantastic cocktail bar – Oriole[1]. Unfortunately they're closed now due to the coronavirus, but once they reopen I highly recommend a visit.

In the meantime if you're into cocktails the bar manager of Nightjar (same owners as Oriole) has some great videos online: https://www.diffordsguide.com/g/1151/tonys-cocktails

I am not affiliated with Oriole or Nightjar in any way, just a huge fan. (So much a fan that I've flown in to the UK on several occasions just to go to these places, can't praise them enough!)

[1]: https://www.oriolebar.com/